\documentclass[letterpaper,11pt]{article}

%=================Begin:Packages===================

\usepackage[utf8]{inputenc}
\usepackage{a4wide}
\usepackage[english]{babel}
\usepackage{scalerel}
\usepackage{hyperref}
\usepackage{amsmath,amsfonts,amsthm} % amssymb
\usepackage{enumitem}
\usepackage{harpoon}
\usepackage[normalem]{ulem}
\usepackage{xargs}                      % Use more than one optional parameter in a new commands
\usepackage[pdftex,dvipsnames]{xcolor}  % Coloured text etc.

%=================End:Packages===================

%==================Begin:Style===================

\newtheorem{definition}{Definition}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}


\newcommandx{\handan}[2]{{\uline{#1}}\textcolor{red}{(#2)}}
%\newcommandx{\handan}[2]{{#1}{}}
\newcommand\mathperiod{.}
\newcommand\mathcomma{,}

%===================End:Style====================

%==================Begin:Macros==================

\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}
\newcommand{\ceil}[1]{\left\lceil #1 \right\rceil}

\newcommand{\vect}[1]{\overrightharp{\ensuremath{#1}}} % {\overset{\rightharpoonup}{#1}}

% probability with square brackets of the right size
% \newcommand{\Pr}{\ensuremath{\mathsf{Pr}}} %probability
% \newcommand{\Prb}[1]{\ensuremath{\Pr\left [#1 \right ]}}

\newcommand*\set[1]{\{ #1 \}} % in text, we don't want {} to grow
\newcommand*\Set[1]{\left\{ #1 \right\}}
\newcommand*\setst[2]{\{ #1 | #2 \}}
\newcommand*\Setst[2]%
        {\left\{\,#1\vphantom{#2} \;\right|\left. #2 \vphantom{#1}\,\right\}}

\newcommand*\List[1]{\left[ #1 \right]}  % cute abbreviations
\newcommand*\listst[2]{[ #1 | #2 ]}
\newcommand*\Listst[2]%
        {\left[\,#1\vphantom{#2} \;\right|\left. #2 \vphantom{#1}\,\right]}

% uniform random selection from set
\newcommand{\randsel}[0]{\ensuremath{\xleftarrow{\text{\$}}}}

% oracles
\newcommand{\ora}[1]{\ensuremath{\mathcal{O}\mathsf{#1}}}
% oracle set
\newcommand{\oraSet}[1]{\ensuremath{\mathcal{O}\textsc{#1}}}
% algorithm
\newcommand{\algo}[1]{\ensuremath{\mathsf{#1}}}
% party
\newcommand{\prt}[1]{\ensuremath{\mathcal{#1}}}
% long-form variable
\newcommand{\var}[1]{\ensuremath{\mathit{#1}}}

%=================End:Macros=====================

%=================Begin:Definitions===================

\newcommand{\VRF}{\algo{VRF}} 
\newcommand{\RingVRF}{\algo{RingVRF}} 
\newcommand{\Sign}{\algo{Sign}} 
\newcommand{\Verify}{\algo{Verify}} 
\newcommand{\KeyGen}{\algo{KeyGen}} 
\newcommand{\Out}{\algo{Out}} 

\newcommand{\sk}{\ensuremath{\mathsf{sk}}} %secret key
\newcommand{\pk}{\ensuremath{\mathsf{pk}}} %public key
\newcommand{\verify}{\ensuremath{\mathsf{Verify}}}
\newcommand{\prove}{\ensuremath{\mathsf{Prove}}}

\newcommand{\vals}{\ensuremath{\mathcal{V}}}
\newcommand{\nvals}{\ensuremath{n}}
\newcommand{\para}{\ensuremath{\rho}}

\newcommand{\pieces}{\ensuremath{\mathcal{D}}}
\newcommand{\prepieces}{\ensuremath{\pieces'}}
\newcommand\merkleroot{\ensuremath{\mathfrak{m}}} % {\mathtt{root}}}  maybe?
\newcommand\receipt{\ensuremath{\mathfrak{C}}}

\newcommand{\val}{\ensuremath{v}}

% \newcommand{\pv}{\ensuremath{\mathcal{P}V}}

\newcommand\grace{\ensuremath{\mathsf{agp}}}

\newcommand{\encode}{\ensuremath{\mathsf{Encode}}}
\newcommand{\decode}{\ensuremath{\mathsf{Decode}}}
% \newcommand{\divi}{\ensuremath{\mathsf{Divide}}}
% \newcommand{\recons}{\ensuremath{\mathsf{Reconstruct}}}

\newcommand{\blobB}{\ensuremath{\bar{B}}} %blob
% \newcommand{\bh}{\ensuremath{B_{head}}} %block header
% \newcommand{\mprf}{\ensuremath{\pi^{mt}}} %merkle proof


\newcommand{\npvals}{\ensuremath{n_0}} % number of parachaim validators
\newcommand\primarychecks{\ensuremath{\mathtt{min_backers}}}   % {\ell_1}
\newcommand\secondarychecks{\ensuremath{\mathtt{min_approvers}}}  % {\ell_2}
\newcommand\totalchecks{\ensuremath{\ell_{\mathrm{TOTALREMOVED}}}} %total number of validity check
\newcommand\equivocationchecks{\ensuremath{\ell_{\mathrm{eqiv}}}}

\newcommand\basesecondarychecks{\ensuremath{\ell_b}} 
\newcommand\rfuv{\ensuremath{\ell_v}} % report factor for unavailability from validators
\newcommand{\runav}{\ensuremath{\ell_a}}
\newcommand{\rinv}{\ensuremath{\ell_f}}

\newcommand\cc{\ensuremath{\mathbin{\|}}}

% \def\newalgo#1{%
% \expandafter\newcommand\csname #1\endcsname{\mathbf{#1}}%
% }
\newcounter{symbols}
\def\newsymbol#1{%
\expandafter\newcommand\csname #1\endcsname{\hyperref[symbols:#1]\mathsf{#1}}%
\refstepcounter{symbols}\label{symbols:#1}\mathsf{#1}%
}


\title{Availability and Validity}


\begin{document}
\date{}
\maketitle

\input{AnV-0-introduction.tex}
\input{AnV-1-sharding.tex}
\input{AnV-2-backing.tex}
\input{AnV-3-availability.tex}
\input{AnV-4-approval.tex}
\input{AnV-5-fishermen.tex}
\input{AnV-6-security.tex}

\appendix
\input{AnV-A1-proof.tex}

\end{document}
